Maddenation

Deal or no Deal

I watched Howie Mandel’s new show the other night and got interested, as usual, in the probability of it all. It’s another million-dollar prize game with an interesting twist. The contestant is presented with 26 briefcases attended by 26 beautiful women. The briefcases contain randomly distributed amounts of money (on printed tags) ranging from $1 million to $0.01, about half of which are less than $1000. The game begins with the contestant picking, but not opening, one of the 26 cases. The probability that s/he has picked a million is obviously1 in 26. That’s where the fun begins.

The contestants are allowed to walk away with what they have, or stay and play the game, which will allow them to better estimate the probability that they have already picked the million-dollar briefcase. They always decide to play.

In the first move, the contestant gets to open 6 other briefcases and record their contents on the big light board behind them. That reduces the number of possibilities to 20 and gives the contestant six amounts that are not in his briefcase. At this point, Howie is interrupted by a phone call from the mysterious “banker” in the isolation booth above the stage. The purpose of this call, Howie tells us, is for the banker to “cut his losses” and make the contestant a monetary offer to quit the game. If the contestant doesn’t accept the offer, then 5 briefcases must be opened on the second move, then 4, then 3, etc. until either all are opened (and the contestant gets what’s in their briefcase) or an offer is accepted. The contestant, in dramatic fashion, must decide if it’s a “Deal or No Deal” to the delight of the audience and those friends and family in attendance. The question for a statistician (like me) is whether or not the probable value of the contestant’s briefcase is more or less than the banker’s offer.

The answer, in theory, is easy. At the start of the game, for example, the distribution of money is such that the average of all 26 amounts is about $131,500. Thus, in theory, if someone were to offer you more than that to let him or her play instead of you, you should take the offer. Opening briefcases allows one to recalculate the average of however many are left, which is the new probable value of the original briefcase. Thus, again in theory, one should only take the banker’s offer if it is greater than the average of the remaining briefcases. Right?

I have only watched the game show a few times, and have only recently been able to list all 26 dollar values contained in the briefcases. However, it seems like the banker’s offer is almost always less than the average of the remaining briefcases. Indeed, if the banker’s intention is the “cut his losses” then his offer will always be less than the computed average. So my burgeoning game strategy is that the contestant should always reject the banker’s offer and play on.

Just tonight, in fact, a woman decided to take an offer (of about $150,000) after a few moves and then found out she had initially picked the million-dollar briefcase! On the other hand, if her briefcase had contained $1, then the offer would look mighty good. So what should one do? Maybe you guys should ask your students. Maybe call the problem “The Statistician’s dilemma.”

Comments

Last night I played the computer version of the game and kept track of probable values to compare with the banker’s offers. The banker was close to the probably value only once, when his offer was about 92% of the probable value. I got all the way down to the last move with the million dollars still in play. The other possibility was $100. So the probable value was a little over half a million. The offer was $355,050, so I said no deal. Then I oppend my suitcase and found the $100. So, in retrospect, I should have taken the 355k$.

When Patrick and I talked about this last week, he made a good point. He said the disappointment at taking home only $100 when you could have had hundreds of thousands is much greater than the disappointment at missing the million. He could be right.